Control Systems

A design-workbench toolkit built around one shared, tested transfer-function engine: enter H(s) once and analyze it everywhere. Poles, zeros, and stability on the s-plane; simulated step, impulse, and ramp responses with the standard performance metrics; exact Bode magnitude and unwrapped phase with gain and phase margins; the Nyquist locus with the algebraic Z = N + P accounting; the full Routh–Hurwitz table with both special cases; and an interactive second-order calculator where sliders on ζ and ωn move the poles and reshape the response live. Every convention — rad/s, negative unity feedback, proper systems, 2% settling — is stated where it is used, because in this subject a silent convention mismatch changes the verdict.

Transfer Function Analyzer

Enter H(s) as polynomial coefficients and get poles, zeros, gain, characteristic equation, system order and type, and a stability verdict — with the pole-zero map and a live step-response preview.

Pole-Zero Analyzer & s-Plane Plot

Poles and zeros on the s-plane with left/right-half-plane classification, the stability verdict, and ζ / ωn of the dominant pole pair.

Step / Impulse / Ramp Response

Simulate the step, impulse, or ramp response of H(s) and read the performance metrics — rise time, peak time, overshoot, settling time — with markers on the curve.

Bode Plot Generator

Magnitude (dB) and unwrapped phase of H(jω) on a log-frequency axis, with gain and phase crossovers located and both stability margins annotated.

Nyquist Plot & Stability Criterion

The L(jω) locus in the complex plane with the −1 point marked, plus the exact Nyquist accounting: open-loop RHP poles P, closed-loop RHP poles Z, and the implied encirclement count N.

Second-Order System Calculator

Sweep ζ and ωn with live sliders and watch the poles move and the step response reshape — with every time- and frequency-domain characteristic computed in closed form.

Stability & Margins Checker

Close the unity-feedback loop around L(s) and check it two independent ways: the Routh–Hurwitz table on the characteristic polynomial, and the gain/phase margins with their crossover frequencies.

PID Controller Analyzer

Tune Kp, Ki, Kd live against your plant: closed-loop step response with metrics, control effort, closed-loop poles, and open-loop margins — all recomputing as you drag.

Ziegler–Nichols PID Tuning

Both classic Ziegler–Nichols methods: closed-loop ultimate gain (Ku, Pu) and open-loop reaction curve (FOPDT), with the tuned gains in both controller forms and a response preview.

Cohen–Coon PID Tuning

Cohen–Coon open-loop tuning from an FOPDT fit (K, T, L) — designed for dead-time-dominant processes — with gains in both forms and a closed-loop preview.

IMC / Lambda PID Tuning

Internal-Model-Control tuning from an FOPDT fit: you choose the closed-loop time constant λ, and the speed-versus-robustness trade-off becomes an explicit engineering decision.

P vs PI vs PID Comparison

One plant, one gain set, three controller structures side by side: what integral action buys (and costs), what derivative action repairs — one chart, one metrics table.

PID Performance Calculator

The complete closed-loop performance report for a tuned PID loop — rise/peak/settling time, overshoot, steady-state error, bandwidth, and margins — exact from the model, or from the second-order approximation, with the validity of that approximation checked.